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• W1837118459 abstract "An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper T> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>T</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a certain type on a space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is presumed to have a branch with some property. It is shown that then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be embedded into a space with an FDD <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper E Subscript i Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(E_i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> so that all normalized sequences in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are almost a skipped blocking of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper E Subscript i Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(E_i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have that property. As an application of our work we prove that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a separable reflexive Banach space and for some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than p greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>1>p>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>C>infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> every weakly null tree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper T> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>T</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the sphere of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a branch <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding=application/x-tex>C</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-equivalent to the unit vector basis of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l Subscript p> <mml:semantics> <mml:msub> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>ell _p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a subspace of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having finite codimension which <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C squared plus epsilon> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>C^2+varepsilon</mml:annotation> </mml:semantics> </mml:math> </inline-formula> embeds into the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script l Subscript p> <mml:semantics> <mml:msub> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>ell _p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> sum of finite dimensional spaces." @default.
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• W1837118459 title "Trees and branches in Banach spaces" @default.
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